Optimizing Cost and Minimizing Energy Loss for Race-Track LHeC Design

1. Optimizing Cost and Minimizing Energy Loss for Race-Track LHeC Design Jake Skrabacz University of Notre Dame Univ. of Michigan CERN REU 2008 CERN AB-department

2. My Problem: Conceptually • LHeC: linear electron collider • Basic Design: linac will be connected to a recirculation track (why?) • Goal: to determine a design for the linac + recirculation structure that will… --Optimize $$$ --Minimize radiative energy loss

3. Primary Considerations in Finding Optimal Design • Cost • Structure (number of accelerations per revolution) • Shape • Size • Number of revolutions • Radiative energy loss

4. Secondary Considerations • Transverse emittance growth from radiation • Number of dipoles needed to keep upper bound on emittance growth • Average length of dipoles • Maximum bending dipole field needed to recirculate beam

5. Primary Shape Studied:The “Race Track” Design 4 Parameters: 1. L: length of linac and/or drift segments, [km] 2. R: radius of bends, [m] 3. bool: boolean (0 for singly-accelerating structure, 1 for doubly-accelerating) 4. N: number of revolutions

6. My Shape Proposal (Rejected):The “Ball Field” Design 5 Parameters: LL: length of linac, [km] LD: length of drift segments, [km] R: small radius, [m] 4. α: angular spread of small circle, [rad] 5. N: number of revolutions

7. My Problem: Analytically Energy Loss to Synchrotron Radiation (around bends): Energy Gain in Linac:

8. My Problem: Computationally(my algorithm) • This optimization problem calls for 8 variables: • 1. Injection energy • 2. Target energy • 3. Energy gradient (energy gain per meter in Linac) • 4. No. of revolutions • 5. bool: singly acc. structure corresponds to 0, while doubly acc. corresponds to 1 • 6. Cost of linac per meter • 7. Cost of drift section per meter • 8. Cost of bending track per meter

9. Algorithm (cont.) • The whole goal is to reduce the cost function to 2 variables—radius and length—then minimize it • Total Cost (R,L) = 2π R N $bend + (1+δ1, bool) L $linac + δ0, boolL $drift • Looking at our structure, and using the energy formulas from the previous slides, you can construct a function that gives the final energy value of the e-beam, E = E (Ei, R, L, dE/dx, revs, bool) • We now have the necessary restriction to our optimization problem: the final energy for the dimensions (R and L) must equal the target energy.

10. The Parameters Used • 1. Injection energy = 500MeV • 2. Target energy = {20, 40, 60, 80, 100, 120} GeV • 3. Energy gradient = 15 MeV/m • 4. No. of revolutions: trials from 1 to 8 • 5. bool: trials with both 0, 1 • 6. Cost of linac per meter = $160k/m • 7. Cost of drift section per meter = $15k/m • 8. Cost of bending track per meter = $50k/m

11. But how do we minimize energy loss? • Create “effective cost,” which incorporates a weight parameter that gives a cost per unit energy loss • Effective Cost = Total Cost + λ ×|ΔErad| • Minimize this!! • Now you have the dual effect: optimize cost and, to the variable extent of the weight parameter, minimize energy loss

12. Conclusions • Reject “ball field” design: reduces energy loss, but cost and size much too large relative to race track!! • Across every target energy and λ value studied, found singly-accelerating structure to be optimal for both total cost and total effective cost • Other optimal parameters (radius, length, number of revolution) depend on target energy and λ value chosen

13. Optimal Cost Results (optimal number of revolutions) Optimal Effective Cost Results

14. Sample ResultE = 80 GeV, λ = $10 million/GeV

15. Limitations • Assumes a constant energy gradient • Assumes cost of bending track independent of size of bend. In reality, the cost of a bending magnet increases with the dipole strength, k ∝ 1/R. • Model does not yet consider lattice structure and the machine’s optics. It gives a “first look” at optimal structure by analyzing macroscopic effects (cost, energy loss, etc). • Model does not yet consider operating cost.

16. Acknowledgements • Univ. of Michigan: Dr. Homer Neal, Dr. Jean Krisch, Dr. Myron Campbell, Dr. Steven Goldfarb • Mentor: Dr. Frank Zimmermann • NSF • CERN

17. Questions?